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Number of Collatz trajectories (A070165) for all positive integers <= 10^n that contain 2^4 as the greatest power of 2 within its trajectory.
1

%I #22 Jul 04 2022 04:38:08

%S 6,89,933,9401,93744,937712,9379078,93773848

%N Number of Collatz trajectories (A070165) for all positive integers <= 10^n that contain 2^4 as the greatest power of 2 within its trajectory.

%C It is conjectured that lim_{n->infinity} a(n)/10^n = 15/16. Empirically, 93.75% of all trajectories have 2^4 as the greatest power of 2 within its trajectory. Sequence A135282(n) is the maximum power of 2 reached in the Collatz trajectory for integer n.

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%e a(1)=6 because the first 10 positive integers have trajectories, of which 6 have 2^4 as the greatest power of 2 in their trajectory.

%e These integers are 3, 5, 6, 7, 9, 10. See trajectory tables below.

%e 1: 1

%e 2: 2 1

%e 3: 3 10 5 16 8 4 2 1

%e 4: 4 2 1

%e 5: 5 16 8 4 2 1

%e 6: 6 3 10 5 16 8 4 2 1

%e 7: 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1

%e 8: 8 4 2 1

%e 9: 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1

%e 10: 10 5 16 8 4 2 1

%t collatz[n_] := Module[{}, If[OddQ[n], 3n+1, n/2]]; step[n_] := Module[{p=0, m=n, q}, While[!IntegerQ[q=Log[2, m]], m=collatz[m]; p++]; {p, q}]; Counts[Table[Last@step[n], {n, 1, 10^5}]][[Key[4]]]

%Y Cf. A070165, A135282.

%K nonn,more

%O 1,1

%A _Frank M Jackson_, Jun 23 2022